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arxiv: 2605.00950 · v2 · submitted 2026-05-01 · 📡 eess.SP · cs.LG· cs.SY· eess.SY

Equation-Free Digital Twins for Nonlinear Structural Dynamics

Mohammad Mahdi Abaei , Ahmad BahooToroody , Arttu Poloj\"arvi , Heikki Remes , Ulf Tyge Tygesen , Mikko Suominen , Michael Beer This is my paper

Pith reviewed 2026-05-12 05:06 UTC · model grok-4.3

classification 📡 eess.SP cs.LGcs.SYeess.SY
keywords digital twinsKoopman operatorstructural dynamicsvirtual sensingoffshore wind turbinedynamic mode decompositionHankel matrixnonlinear systems
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The pith

Lifting sensor data into a linear invariant subspace reconstructs structural states without any prior mass or stiffness matrices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a data-driven digital twin method for high-dimensional structures facing nonlinear kinematics and unknown excitations where conventional models break down. By embedding measurements through a rank-optimized Koopman-Hankel construction, the approach creates an autonomous linear representation that supports real-time state reconstruction and virtual sensing from partial data. This matters for systems like floating offshore wind turbines because it removes the need to supply detailed physical parameters upfront while still separating true resonances from harmonics and noise. Validation on the NREL 5MW spar-buoy turbine shows R-squared values above 0.95 at 1 Hz assimilation rates, along with an estimated predictability limit of roughly one second.

Core claim

The central claim is that a rank-optimized Koopman-Hankel manifold formed from operational time-series data produces a linear invariant subspace in which the dynamics of a nonlinear structural system can be reconstructed autonomously and without knowledge of inputs or a priori mass and stiffness matrices, achieving reconstruction accuracy greater than 0.95 at 1 Hz data assimilation on the NREL 5MW spar-buoy turbine while separating structural resonances from deterministic 3P harmonics under colored noise.

What carries the argument

The rank-optimized Koopman-Hankel manifold, an embedding of measured time histories into a higher-dimensional space where the evolution operator appears linear, which is then decomposed to extract modes and enable input-blind reconstruction.

If this is right

  • Real-time virtual sensing at structural hotspots becomes feasible even when only partial sensor data is available.
  • The method quantifies a physical predictability horizon of about 1 second from the estimated Lyapunov time.
  • Reconstruction works for coupled aero-hydro-servo-elastic systems without requiring full physics models.
  • Standard subspace identification methods are outperformed when deterministic harmonics and colored noise are present.
  • Autonomous identification continues under sensor failure or non-stationary forcing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting approach could be tested on experimental rather than simulated data to confirm mode separation under real measurement conditions.
  • Extending the framework to other large-scale structures such as bridges or aircraft wings might reduce reliance on detailed finite-element models for ongoing monitoring.
  • The short predictability horizon suggests that any digital twin of this type will require frequent measurement updates to stay accurate beyond one second.
  • Combining the method with existing sparse sensor networks could lower the instrumentation density needed in remote or extreme environments.

Load-bearing premise

That choosing the embedding rank will reliably isolate structural resonances from deterministic harmonics and colored noise without case-specific tuning that alters the reported reconstruction accuracy.

What would settle it

Applying the fixed-rank procedure to a new structure whose known natural frequencies differ from those recovered on the NREL turbine and checking whether the R-squared at 1 Hz assimilation drops below 0.9 or the separation from harmonics fails.

Figures

Figures reproduced from arXiv: 2605.00950 by Ahmad BahooToroody, Arttu Poloj\"arvi, Heikki Remes, Michael Beer, Mikko Suominen, Mohammad Mahdi Abaei, Ulf Tyge Tygesen.

Figure 1
Figure 1. Figure 1: Schematic of the Spar FOWT. The diagram illustrates the coupled aero-hydro-servo view at source ↗
Figure 2
Figure 2. Figure 2: The evolution of states x in the vector space M (top layer) and observables g(x) in the functional space (bottom layer). The Koopman operator K enables linear evolution of the observables, bypassing the nonlinear map f. encodes the trajectory history. Let yk ∈ R p denote the multivariate state vector at time step k, containing measurements from all p sensors simultaneously. For a system with p sensors and … view at source ↗
Figure 3
Figure 3. Figure 3: Extraction of spatio-temporal coherent structures from wind turbine sensor data via view at source ↗
Figure 4
Figure 4. Figure 4: Schematic of the Data-Driven Virtual Sensing framework, detailing the data flow from view at source ↗
Figure 5
Figure 5. Figure 5: Probabilistic operational mode shapes derived from the ensemble of load cases and view at source ↗
Figure 6
Figure 6. Figure 6: Reconstructed time-domain evolution of the identified structural modes are shown for view at source ↗
Figure 7
Figure 7. Figure 7: Discrete eigenvalue spectra of the identified system. The structural modes (red stars) view at source ↗
Figure 8
Figure 8. Figure 8: Time-series reconstruction results for the 1.0 s update horizon. Top row compares a view at source ↗
Figure 9
Figure 9. Figure 9: Long-term reconstruction fidelity for Sensor 8, representing nacelle acceleration over view at source ↗
Figure 10
Figure 10. Figure 10: Singular value spectrum (σk) of the Hankel matrix illustrating the energy distribution across dynamic modes. The spectrum is divided into two operational regimes: Zone I (24 ≤ k ≤ 34) represents the high-energy subspace governing global structural modes, suitable for Operational Modal Analysis. Zone II (50 ≤ k ≤ 90) highlights the sub-dominant, low-energy subspace required for high-fidelity Virtual Sensin… view at source ↗
Figure 11
Figure 11. Figure 11: Estimation of the maximal Lyapunov exponent ( view at source ↗
read the original abstract

Monitoring high-dimensional engineering structures in extreme environments is limited by non-stationary excitation, nonlinear structural kinematics, and stochastic forcing. Traditional model-based and black-box data-driven methods often struggle to resolve these dynamics in real time, particularly under sensor failure or partial observability. This paper introduces a rank-optimized digital twin framework based on Koopman operator theory, Hankel-matrix embeddings, and dynamic mode decomposition. By lifting operational data into a linear invariant subspace, the method enables autonomous, input-blind reconstruction of structural states without requiring a priori mass or stiffness matrices. The framework is validated on an NREL 5MW spar-buoy floating offshore wind turbine, representing a challenging coupled aero-hydro-servo-elastic system. Results show that the rank-optimized Koopman-Hankel manifold separates structural resonances from deterministic 3P rotor harmonics under colored noise, where standard subspace identification can be unreliable. A rolling-horizon virtual sensing strategy achieves high-fidelity reconstruction at critical structural hotspots, with coefficient of determination greater than 0.95 at 1 Hz data assimilation and accuracy exceeding 0.99 at higher sampling rates. By estimating a physical Lyapunov time of approximately 1.0 s, the study defines the predictability horizon associated with the system information barrier. The proposed framework provides a computationally efficient and resilient digital twin approach for real-time identification and virtual sensing of complex structural dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an equation-free digital twin framework for nonlinear structural dynamics using rank-optimized Koopman operator theory combined with Hankel-matrix embeddings and dynamic mode decomposition. The approach lifts operational data into a linear invariant subspace to enable autonomous, input-blind reconstruction of structural states without a priori mass or stiffness matrices. Validation on the NREL 5MW spar-buoy floating offshore wind turbine demonstrates separation of structural resonances from 3P harmonics under colored noise, achieving R² > 0.95 at 1 Hz data assimilation and >0.99 at higher rates, with an estimated Lyapunov time of ~1 s defining the predictability horizon.

Significance. If the central claims hold after clarification, the work offers a computationally efficient and resilient approach to real-time virtual sensing for high-dimensional nonlinear systems under stochastic forcing, addressing limitations of traditional model-based methods in extreme environments like offshore wind energy. The explicit link to a physical Lyapunov time as an information barrier provides a quantifiable predictability metric that could inform operational decisions in structural health monitoring.

major comments (2)
  1. [Abstract] Abstract, second paragraph: The load-bearing claim of 'autonomous, input-blind reconstruction ... without requiring a priori mass or stiffness matrices' rests on the rank-optimized Koopman-Hankel manifold cleanly separating structural resonances from 3P harmonics and colored noise. No criterion for manifold rank selection (singular-value threshold, reconstruction error, or otherwise) is stated; if this choice is evaluated on the same time series used to report R² > 0.95, the separation is no longer fully autonomous and the central contribution is undermined.
  2. [Validation on NREL 5MW] Validation on NREL 5MW spar-buoy (results paragraph): The reported fidelity (R² > 0.95 at 1 Hz assimilation, Lyapunov time ~1.0 s) and superiority over standard subspace identification are presented without baseline quantitative comparisons, data-exclusion rules, or error-propagation analysis. This absence prevents verification that the quoted performance is independent of rank tuning and directly affects the strength of the 'resilient digital twin' conclusion.
minor comments (2)
  1. [Abstract] Abstract: 'accuracy exceeding 0.99 at higher sampling rates' is imprecise; state the exact rates and confirm whether the metric is R² or another quantity.
  2. Throughout: Define acronyms (DMD, NREL, etc.) at first use and ensure consistent notation for the Koopman-Hankel manifold across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have prompted us to strengthen the clarity and rigor of the presentation. We address each major comment below and have incorporated revisions to the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract, second paragraph: The load-bearing claim of 'autonomous, input-blind reconstruction ... without requiring a priori mass or stiffness matrices' rests on the rank-optimized Koopman-Hankel manifold cleanly separating structural resonances from 3P harmonics and colored noise. No criterion for manifold rank selection (singular-value threshold, reconstruction error, or otherwise) is stated; if this choice is evaluated on the same time series used to report R² > 0.95, the separation is no longer fully autonomous and the central contribution is undermined.

    Authors: We agree that an explicit statement of the rank-selection criterion is necessary to substantiate the autonomy claim. The manuscript applies a fixed singular-value energy threshold (retaining 95% of the cumulative Hankel-matrix energy) that is chosen once from standard DMD practice and applied uniformly; this threshold is independent of the reported R² values. To eliminate any ambiguity, we have revised the abstract to mention the energy-threshold criterion and added a dedicated paragraph in the Methods section that defines the threshold mathematically and demonstrates its application on separate training segments. These changes preserve the input-blind character of the procedure while directly addressing the referee's concern. revision: yes

  2. Referee: [Validation on NREL 5MW] Validation on NREL 5MW spar-buoy (results paragraph): The reported fidelity (R² > 0.95 at 1 Hz assimilation, Lyapunov time ~1.0 s) and superiority over standard subspace identification are presented without baseline quantitative comparisons, data-exclusion rules, or error-propagation analysis. This absence prevents verification that the quoted performance is independent of rank tuning and directly affects the strength of the 'resilient digital twin' conclusion.

    Authors: We acknowledge that the original results section relied primarily on qualitative statements of superiority. In the revised manuscript we have added a quantitative comparison table that reports R² and RMSE for the proposed method against N4SID and unoptimized DMD on identical NREL 5MW datasets. We have also inserted a short subsection describing data-exclusion rules (removal of segments with >10% missing samples or outlier variance exceeding three standard deviations) and a first-order error-propagation estimate linking DMD eigenvalue uncertainty to the Lyapunov-time calculation. These additions confirm that the quoted metrics remain stable for rank choices within the 95% energy band and thereby support the resilience claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The provided abstract and method description define a rank-optimized Koopman-Hankel DMD pipeline whose central outputs (subspace reconstruction, R² metrics, Lyapunov time) are presented as computed results from operational data. No equations or steps are shown that define the reported accuracy or separation performance as an input to rank selection, nor is any load-bearing premise reduced to a self-citation or fitted parameter renamed as prediction. Validation metrics are reported as independent outcomes of the autonomous procedure on the NREL turbine dataset. This satisfies the default expectation of non-circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract does not enumerate explicit free parameters or invented entities; the framework rests on standard Koopman operator assumptions for nonlinear systems.

free parameters (1)
  • manifold rank
    The optimized rank used to separate resonances from harmonics is chosen or fitted during the procedure.
axioms (1)
  • domain assumption Operational data can be lifted into a linear invariant subspace that captures the underlying nonlinear dynamics
    Central premise of the Koopman-Hankel approach stated in the abstract.

pith-pipeline@v0.9.0 · 5581 in / 1290 out tokens · 34416 ms · 2026-05-12T05:06:55.274723+00:00 · methodology

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Reference graph

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